wait, i thought i read in here a while ago that projectiles were half ovals... is that right? i mean, i know we all just do the math as if it were a parabola, but its actually half an oval because of the curvature of the earth right? or did i totally just make that up...

Originally posted by Gale17 wait, i thought i read in here a while ago that projectiles were half ovals... is that right? i mean, i know we all just do the math as if it were a parabola, but its actually half an oval because of the curvature of the earth right? or did i totally just make that up...

I think I know what you're thinking, but you have it a little garbled.

First, by "oval" I'm guessing you mean an ellipse. It's true that if you throw an object, it follows an elliptical orbit (if you throw it at less than escape velocity). A partial arc of an ellipse is not a parabola. (i.e., if you cut a piece off an ellipse, the piece is never a parabola.) However, over a small distance, an arc of an ellipse is well approximated by a parabola -- they're very similar in shape.

So, we say that if you throw a ball, it follows a parabolic arc. But it's really following part of an elliptical arc that, over a comparatively small distance, is very similar to (but not the same as) a parabola.

yeah thanks Ambitwistor that's it... heh... i actually sat for like 5 minutes trying to figure out the other word for oval... but its late, didn't come to me... but yeah, i knew they were close anyways so it didn't really matter... but i was just making sure that i hadn't made something up...

Staff: Mentor

Originally posted by Ambitwistor It's true that if you throw an object, it follows an elliptical orbit (if you throw it at less than escape velocity). A partial arc of an ellipse is not a parabola. (i.e., if you cut a piece off an ellipse, the piece is never a parabola.) However, over a small distance, an arc of an ellipse is well approximated by a parabola -- they're very similar in shape.

Another way to think of it is circles and parabolas are special cases of ellipses. A circle is an ellipse with zero distance between the foci and a parabola is an ellipse with infinite distance between the foci. That is why over small distances a parabola and ellipse are very close - they are siblings (conic sections).

Originally posted by russ_watters Another way to think of it is circles and parabolas are special cases of ellipses. A circle is an ellipse with zero distance between the foci and a parabola is an ellipse with infinite distance between the foci. That is why over small distances a parabola and ellipse are very close - they are siblings (conic sections).

Not really. It's true that parabolas and ellipses are conic sections with the properties you note. But the reason why a small arc of an ellipse looks like a parabola has nothing to do with the fact that they are both conic sections.

In fact, any trajectory due to any force law -- not just conic section solutions to an inverse square law -- will look like a parabola over small distances. This is a consequence of Taylor's theorem, expanding to second order. It arises because any gravitational field locally looks like a uniform field over a distance scale smaller than the scale of its gradient, and a parabola is the trajectory obtained from a uniform field.

Projectile motion is always, ultimately, elliptical. FOr the motion to be truly parabolic, the acceleration due to gravity would have to always be in one direction. THis condition is only approximated when we are near the earth's surface because we cannot detect the change in direction of "g". Over relatively small distances we use parabolic projectile motion. FOr intercontinental ballistic missiles, however, elliptical projectile motion is followed (plus coriolis effect, air resistance, and other fun stuff, and of course for satellite motion, it's totally elliptical.